Commutator subspace
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In mathematics, the commutator subspace of a two-sided
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
of bounded linear operators on a separable
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
is the linear subspace spanned by commutators of operators in the ideal with bounded operators. Modern characterisation of the commutator subspace is through the
Calkin correspondence In mathematics, the Calkin correspondence, named after mathematician John Williams Calkin, is a bijective correspondence between two-sided ideals of bounded linear operators of a separable infinite-dimensional Hilbert space and Calkin sequence space ...
and it involves the invariance of the Calkin sequence space of an operator ideal to taking Cesàro means. This explicit spectral characterisation reduces problems and questions about commutators and
traces Traces may refer to: Literature * ''Traces'' (book), a 1998 short-story collection by Stephen Baxter * ''Traces'' series, a series of novels by Malcolm Rose Music Albums * ''Traces'' (Classics IV album) or the title song (see below), 1969 * ''Tra ...
on two-sided ideals to (more resolvable) problems and conditions on sequence spaces.


History

Commutators of linear operators on Hilbert spaces came to prominence in the 1930s as they featured in the
matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum j ...
, or Heisenberg, formulation of quantum mechanics. Commutator subspaces, though, received sparse attention until the 1970s. American mathematician
Paul Halmos Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...
in 1954 showed that every bounded operator on a separable infinite dimensional Hilbert space is the sum of two commutators of bounded operators. In 1971 Carl Pearcy and David Topping revisited the topic and studied commutator subspaces for Schatten ideals. As a student American mathematician Gary Weiss began to investigate spectral conditions for commutators of Hilbert–Schmidt operators. British mathematician
Nigel Kalton Nigel John Kalton (June 20, 1946 – August 31, 2010) was a British-American mathematician, known for his contributions to functional analysis. Career Kalton was born in Bromley and educated at Dulwich College, where he excelled at both ma ...
, noticing the spectral condition of Weiss, characterised all trace class commutators. Kalton's result forms the basis for the modern characterisation of the commutator subspace. In 2004 Ken Dykema,
Tadeusz Figiel Tadeusz Figiel (born 2 July 1948 in Gdańsk) is a Polish mathematician specializing in functional analysis. Biography In 1970 Figiel graduated in mathematics at the University of Warsaw. He received his doctorate in 1972 under the supervision ...
, Gary Weiss and
Mariusz Wodzicki Mariusz Wodzicki (Count Wodzicki) (born 1956) is a Polish mathematician and nobleman, whose works primarily focus on analysis, algebraic k-theory, noncommutative geometry, and algebraic geometry. Wodzicki was born in Bytom, Poland in 1956. He rec ...
published the spectral characterisation of normal operators in the commutator subspace for every two-sided ideal of compact operators.


Definition

The commutator subspace of a two-sided ideal ''J'' of the bounded linear operators ''B''(''H'') on a separable Hilbert space ''H'' is the linear span of operators in ''J'' of the form 'A'',''B''nbsp;= ''AB'' − ''BA'' for all operators ''A'' from ''J'' and ''B'' from ''B''(''H''). The commutator subspace of ''J'' is a linear subspace of ''J'' denoted by Com(''J'') or 'B''(''H''),''J''


Spectral characterisation

The
Calkin correspondence In mathematics, the Calkin correspondence, named after mathematician John Williams Calkin, is a bijective correspondence between two-sided ideals of bounded linear operators of a separable infinite-dimensional Hilbert space and Calkin sequence space ...
states that a
compact operator In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact c ...
''A'' belongs to a two-sided ideal ''J'' if and only if the
singular values In mathematics, in particular functional analysis, the singular values, or ''s''-numbers of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the self- ...
μ(''A'') of ''A'' belongs to the Calkin sequence space ''j'' associated to ''J''.
Normal operator In mathematics, especially functional analysis, a normal operator on a complex Hilbert space ''H'' is a continuous linear operator ''N'' : ''H'' → ''H'' that commutes with its hermitian adjoint ''N*'', that is: ''NN*'' = ''N*N''. Normal operat ...
s that belong to the commutator subspace Com(''J'') can characterised as those ''A'' such that μ(''A'') belongs to ''j'' ''and'' the Cesàro mean of the sequence μ(''A'') belongs to ''j''. The following theorem is a slight extension to differences of normal operators (setting ''B''  0 in the following gives the statement of the previous sentence). : Theorem. Suppose ''A,B'' are compact normal operators that belong to a two-sided ideal ''J''. Then ''A'' − ''B'' belongs to the commutator subspace Com(''J'') if and only if :::: \left\_^\infty \in j :where ''j'' is the Calkin sequence space corresponding to ''J'' and ''μ''(''A''), ''μ''(''B'') are the singular values of ''A'' and ''B'', respectively. Provided that the eigenvalue sequences of all operators in ''J'' belong to the Calkin sequence space ''j'' there is a spectral characterisation for arbitrary (non-normal) operators. It is not valid for every two-sided ideal but necessary and sufficient conditions are known. Nigel Kalton and American mathematician Ken Dykema introduced the condition first for countably generated ideals. Uzbek and Australian mathematicians Fedor Sukochev and Dmitriy Zanin completed the eigenvalue characterisation. : Theorem. Suppose ''J'' is a two-sided ideal such that a bounded operator ''A'' belongs to ''J'' whenever there is a bounded operator ''B'' in ''J'' such that :If the bounded operator ''A'' and ''B'' belong to ''J'' then ''A'' − ''B'' belongs to the commutator subspace Com(''J'') if and only if :::: \left\_^\infty \in j :where ''j'' is the Calkin sequence space corresponding to ''J'' and ''λ''(''A''), ''λ''(''B'') are the sequence of eigenvalues of the operators ''A'' and ''B'', respectively, rearranged so that the absolute value of the eigenvalues is decreasing. Most two-sided ideals satisfy the condition in the Theorem, included all Banach ideals and quasi-Banach ideals.


Consequences of the characterisation

* Every operator in ''J'' is a sum of commutators if and only if the corresponding Calkin sequence space ''j'' is invariant under taking Cesàro means. In symbols, Com(''J'')  ''J'' is equivalent to C(''j'')  ''j'', where C denotes the Cesàro operator on sequences. * In any two-sided ideal the difference between a positive operator and its diagonalisation is a sum of commutators. That is, ''A'' − diag(''μ''(''A'')) belongs to Com(''J'') for every positive operator ''A'' in ''J'' where diag(''μ''(''A'')) is the diagonalisation of ''A'' in an arbitrary orthonormal basis of the separable Hilbert space ''H''. * In any two-sided ideal satisfying () the difference between an arbitrary operator and its diagonalisation is a sum of commutators. That is, ''A'' − diag(''λ''(''A'')) belongs to Com(''J'') for every operator ''A'' in ''J'' where diag(''λ''(''A'')) is the diagonalisation of ''A'' in an arbitrary orthonormal basis of the separable Hilbert space ''H'' and ''λ''(''A'') is an eigenvalue sequence. * Every quasi-nilpotent operator in a two-sided ideal satisfying () is a sum of commutators.


Application to traces

A trace φ on a two-sided ideal ''J'' of ''B''(''H)'' is a linear functional φ:''J'' → \mathbb that vanishes on Com(''J''). The consequences above imply * The two-sided ideal ''J'' has a non-zero trace if and only if C(''j'') ≠ ''j''. * ''φ''(''A'') = ''φ'' ∘ diag(''μ''(''A'')) for every positive operator ''A'' in ''J'' where diag(''μ''(''A'')) is the diagonalisation of ''A'' in an arbitrary orthonormal basis of the separable Hilbert space ''H''. That is, traces on ''J'' are in direct correspondence with
symmetric functional Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
s on ''j''. * In any two-sided ideal satisfying (), ''φ''(''A'') = ''φ'' ∘ diag(''λ''(''A'')) for every operator ''A'' in ''J'' where diag(''λ''(''A'')) is the diagonalisation of ''A'' in an arbitrary orthonormal basis of the separable Hilbert space ''H'' and ''λ''(''A'') is an eigenvalue sequence. * In any two-sided ideal satisfying (), ''φ''(''Q'') = 0 for every quasi-nilpotent operator ''Q'' from ''J'' and every trace ''φ'' on ''J''.


Examples

Suppose ''H'' is a separable infinite dimensional Hilbert space. * Compact operators. The compact linear operators ''K''(''H'') correspond to the space of converging to zero sequences, ''c''0. For a converging to zero sequence the Cesàro means converge to zero. Therefore, C(''c''0) = ''c''0 and Com(''K''(''H''))  ''K''(''H''). * Finite rank operators. The finite rank operators ''F''(''H'') correspond to the space of sequences with finite non-zero terms, ''c''00. The condition :::: \left\_^\infty \in c_ :occurs if and only if :::: a_1 + a_2 + \cdots + a_N = 0 :for the sequence (''a''1, ''a''2, ... , ''a''''N'', 0, 0 , ...) in ''c''00. The kernel of the operator trace Tr on ''F''(''H'') and the commutator subspace of the finite rank operators are equal, ker Tr  Com(''F''(''H'')) ⊊ ''F''(''H''). * Trace class operators. The
trace class operator In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace ...
s ''L''1 correspond to the summable sequences. The condition :::: \left\_^\infty \in \ell_ :is stronger than the condition that ''a''1 + ''a''2 ... = 0. An example is the sequence with :::: a_n = \frac , \quad n \geq 2 . :and :::: a_1 = - \sum_^\infty a_n. which has sum zero but does not have a summable sequence of Cesàro means. Hence Com(''L''1) ⊊ ker Tr ⊊ ''L''1. * Weak trace class operators. The weak trace class operators ''L''1,∞ correspond to the weak-''l''1 sequence space. From the condition :::: \left\_^\infty \in \ell_ :or equivalently :::: \left\_^\infty = O(1) it is immediate that Com(''L''''1'',∞)+  (''L''1)+. The commutator subspace of the weak trace class operators contains the trace class operators. The harmonic sequence 1,1/2,1/3,...,1/''n'',... belongs to ''l''1,∞ and it has a divergent series, and therefore the Cesàro means of the harmonic sequence do not belong to ''l''1,∞. In summary, ''L''1 ⊊ Com(''L''1,∞) ⊊ ''L''1,∞.


Notes


References

* * * * {{cite book , isbn=978-3-11-026255-1 , author= S. Lord, F. A. Sukochev. D. Zanin , year=2012 , url=http://www.degruyter.com/view/product/177778 , title=Singular traces: theory and applications , publisher=De Gruyter , location=Berlin , doi= 10.1515/9783110262551 Hilbert spaces Von Neumann algebras